Pressure sensitive optical fiber having optimized coatings

ABSTRACT

The acoustic pressure sensitivity of an optical fiber waveguide is maximized by surrounding it with an outer coating which is selected from a material having high Young&#39;s modulus and low bulk modulus.

FIELD OF THE INVENTION

This invention relates to coatings for optical fibers to maximize theiracoustic sensitivity and more particularly to such coatings which areselected from materials having high Young's modulus and/or low bulkmodulus.

BACKGROUND DESCRIPTION

Phase modulation in single-mode optical fibers has been successfullyused as a transduction mechanism for detecting environmental changessuch as pressure, acoustic fields, magnetic and electric fields, andtemperature. Mach-Zehnder interferometric fiber-optic sensors have beenstudied analytically and have been tested both in the laboratory and inreal environments and demonstrate very promising performance.

With an interferometric fiber optic sensor as shown in FIG. 1, optimumperformance is achieved with maximum signal to noise ratio which signalis obtained by maximizing the sensitivity of the sensing fiber anddesensitizing the sensitivity of the lead and reference fibers. U.S.Pat. No. 4,427,263 issued to Lagakos et al. on Jan. 24, 1984, disclosespressure insensitive optical fibers for use in the reference and leadarms of the interferometer.

The acoustic sensitivity of an optical fiber can be maximized byappropriately selecting the fiber composition and design, as will bedisclosed.

The pressure sensitivity of the optical phase in a fiber is defined asthe magnitude of Δφ/φΔP, where Δφ is the shift in the phase delay φ dueto a pressure change ΔP. If a given pressure change, ΔP, results in afiber core axial strain ε_(z) and radial strain ε_(r), then it can beshown that: ##EQU1## Here P₁₁ and P₁₂ are the elasto-optic coefficientsof the core and n is the refractive index of the core. The first term inequation (1) is the part of Δφ/φΔP which is due to the fiber lengthchange, while the second and third terms are the parts due to therefractive index modulation of the core, which is related to thephotoelastic effect. Stated another way Δφ/φΔP=the algebraic sum of thephase change due to the fiber length change plus phase change due torefractive index change. The objective is to maximize the fibersensitivity Δφ/φΔP.

A typical optical fiber (FIG. 2a) is composed of a core, cladding, and asubstrate from glasses having similar properties. This glass fiber isusually coated with a soft rubber and then with a hard plastic. In orderto calculate the sensitivity as given in Eq.(1) the strains in the coreε_(z) and ε_(r) must be related to properties of the various layer ofthe fiber. The strains ε_(z) and ε_(r), which are related to thegeometry and composition of the fiber, can be calculated from thestresses and displacements in the various fiber layers by applying theappropriate boundary conditions. In this analysis we have taken intoaccount the exact geometry of a typical four layer fiber as shown inFIG. 2a.

The polar stresses τ_(r), τ.sub.θ, and τ_(z) in the fiber are related tothe strains ε_(r), ε.sub.θ, and ε_(z) as follows: ##EQU2## where i isthe layer index (0 for the core, 1 for the cladding, etc.) and λ^(i) andμ^(i) are the Lame parameters which are related to the Young's modulus,E^(i), and Poisson's ratio, V^(i), as follows: ##EQU3##

For a cylinder the strains can be obtained from the Lame solutions##EQU4## where U_(o) ^(i), U_(l) ^(i), and W_(o) ^(i) are constants tobe determined. Since the strains must be finite at the center of thecore, U_(l) ^(o) =0.

For a fiber with m layers, the constants U_(o) ^(i), U_(l) ^(i), andW_(o) ^(i) in Eq. (4) are determined from the boundary conditions:

    τ.sub.r.sup.i |.sub.r=ri =τ.sub.r.sup.i+1 |.sub.r=ri,(i=0,1, . . . . , m-1)                (5)

    μ.sub.r.sup.i |.sub.r=ri =μ.sub.r.sup.i+1 |.sub.r=ri,(i=0,1, . . . . , m-1)                (6)

    τ.sub.r.sup.m |.sub.r=rm =-P,                 (7) ##EQU5##

    ε.sub.z.sup.o =ε.sub.z.sup.1 =. . . =ε.sub.z.sup.m( 9)

where u_(r) ^(j) (=∫ε_(r) ^(i) dr) is the radial displacement in thei^(th) layer, and r_(i) and A_(i) are the radius and the cross sectionarea of the i^(th) layer respectively. Equations (5) and (6) describethe radial stress and displacement continuity across the boundaries ofthe layers. Equations (7) and (8) assume that the applied pressure ishydrostatic. Equation (9) is the plane strain approximation whichignores end effects. For long thin cylinders, such as fibers, itintroduces an error of less than 1%. Using the boundary conditionsdescribed by Eqs. (5)-(9), the constants U_(o) ^(i), U_(l) ^(i), andW_(o) ^(i) are determined and ε_(r) ^(o) and ε_(z) ^(o) are calculatedfrom Eq.(4). Then, from Eq.(1) the sensitivity, Δφ/ΔφP, can be found.

In the past no one has recognized the relationship between the bulkmodulus, Young's modulus of the fiber coating, the fiber thickness, andthe the resultant acoustic pressure sensitivity of the optical fiber.

SUMMARY OF THE INVENTION

Accordingly, it is a general object of the invention to provide apressure sensitive optical fiber waveguide having optimum sensitivity.

It is a more specific object of the invention to optimize the opticalfiber waveguide pressure sensitivity per given area of fiber.

It is a further object of the invention to optimize the optical fiberwaveguide pressure sensitivity in the sensor arm of an interferometricacoustic sensor.

These and other objects of the invention are accomplished by having apressure sensitive optical fiber which is coated with materials havinghigh Young's modulus and/or low bulk modulus. Once the Young's and bulkmodulus of a coating material are established, the acoustic pressuresensity of that coated fiber can be calculated.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same become betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is an illustration of a Mach-Zehnder optical fiber interferometerand leads thereto for sensing external acoustic pressure;

FIG. 2 is an optical fiber with illustrated acoustic pressures whichinduce phase shifts;

FIG. 2a is a cross-sectional view of fiber sensor prepared according tothe invention;

FIG. 3 illustrates the acoustic sensitivity of a fiber as a function ofcoating thickness for different coating Young's moduli with a constantbulk moduli;

FIG. 4 illustrates the acoustic sensitivity of a fiber as a function ofbulk modulus and Young's modulus; and

FIG. 5 illustrates the acoustic sensitivity per fiber cross-section vsfiber coating Young's modulus and outer diameter.

DESCRIPTION OF THE PREFERRED EMBODIMENT

As in most interferometric sensors, a Mach-Zehnder arrangement was used(shown schematically in FIG. 1). FIG. 1 shows an apparatus in the formof a Mach-Zehnder optical fiber interferometer 10 for sensing acousticwave pressure. Optical fibers 12 and 14, making up the sensing andreference arms, respectively, are relatively short lines, usually just afew meters or less. The sensing arm is obviously exposed to theenvironment for encountering the conditions it is designed to sense. Thereference arm may be isolated or shielded by the environment (means notillustrated) or since it is relatively short, it may be exposed to theenvironment along with the sensing arm. This, of course, depends uponits type of service.

The optical fiber interferometer performs in the usual manner. A lightsource 18 introduces light into lead optical fiber 16 where it travelsand is subsequently divided to enter parallel arms 12 and 14. The lightpasses through the arms where a relative phase shift is induced in thesensing arm by a copndition such as a magnetic field or acoustic wave.The light is then recombined and passed through lead 16' back todetector 20 where the extent of induced phase shifts is detected andconverted into a meaningful reading which is in proportion to themagnitude of the conditions causing the phase shift. If there is anacousto-optic non-reciprocity between these two fiber arms, which is tosay a difference in the acoustically induced phase shift (pressuresensitivity), Δφ/φΔP, then acoustic pressures can be sensed with aminimum detectable pressure P_(min), as: ##EQU6##

FIG. 2 is a representation of an optical fiber in an acoustic conditionwherein pressures thereon cause dimension changes which result in bothchanges in optical path length and changes in the refractive index aspreviously explained under "Background of the Invention" heading. FIG.2a illustrates the different layers in a typical fiber. However, thesubstrate layer c and the soft first coating layer d are not necessaryfor this invention.

In this study, the acoustic response of coated fibers has been examinedanalytically. The coating material parameters necessary to do this,e.g., the elastic moduli, have been determined experimentally byappropriate measurements in bulk samples for the various coatingmaterials.

In general, the acoustic sensitivity is a very strong function of theelastic moduli of the outer hard polymeric coating of the fiber. Highsensitivity can be achieved with coatings of low bulk modulus and highYoung's modulus materials. This can be understood as follows. FIG. 3shows the acoustic sensitivity of a fiber as a function of coatingthickness for different coating Young's moduli. All the other parametersof the fiber were kept constant and are listed in Table I. As can beseen from this figure, for thick coatings (somewhat greater than 0.5 mm)the sensitivity is determined by the bulk modulus which governs thefiber dimensional changes. For typical fibers (approx. 0.5 mmthickness), however, both the bulk modulus and the Young's modulus areimportant.

                  TABLE I                                                         ______________________________________                                        STANDARD ITT SINGLE-MODE FIBER                                                (WITHOUT OUTER COATING)                                                                                 Sub-    First Coating                                       Core     Cladding strate  (Soft)                                      ______________________________________                                        Composition                                                                             SiO.sub.2 + traces                                                                       95% SiO.sub.2                                                                          SiO.sub.2                                                                           Silicone                                            of GeO.sub.2 (0.1%)                                                 Diameter (μm)                                                                         4         26       84    250                                       Young's   72         65       72    0.0035                                    Modulus                                                                       (10.sup.10 dyne/cm.sup.2)                                                     Poisson's Ratio                                                                         0.17       0.149    0.17  0.49947                                   P.sub.11  0.126                                                               P.sub.12  0.27                                                                n         1.458                                                               ______________________________________                                    

FIG. 4 is a plot of pressure sensitivity vs bulk modulus and Young'smodulus. The solid lines in the plot of FIG. 4 indicate the relationshipbetween these three parameters which were derived analytically (as wasFIG. 3) from the equations given in the "Background of the Invention".As can be seen from FIG. 4, high sensitivity requires low bulk modulusand high Young's modulus coatings. In this case, the bulk modulusdetermines the "maximum" fiber dimensional changes, while the Young'smodulus governs the fraction of these changes, or strains, which cancommunicate to the fiber core. Even for small coating thickness (muchless than 0.5 mm) a high Young's modulus results in high sensitivitysince it can communicate a high fraction of the applied stress to thefiber core, while low Young's modulus requires substantial coatingthickness. The dotted area in FIG. 4 indicates materials which wereactually found during the experimental investigation as given below indetail in Tables II-V.

The important role of the Young's modulus is also apparent from FIG. 5where Δφ/φΔP is shown as a function of thickness and Young's modulus ofthe coating. The bulk modulus was kept constant (2.5×10¹⁰ dyn/cm₂).

Thus in applications where the size of the sensor limits the maximumlength of the sensing fiber the sensitivity per cross section should bemaximized. This sensitivity is shown as a function of thickness andYoung's modulus of the coating for constant bulk modulus (2.5×10¹⁰dyn/cm²). From this figure it can be seen that the sensitivity per crosssection (a) increases with increasing Young's modulus and (b) ismaximized with a Young's modulus dependent on coating thickness. Evenfor small coating thickness a high Young's modulus results in highsensitivity since it can communicate high fraction of the applied stressto the fiber core, while low Young's modulus requires substantialcoating thickness. Therefore, in designing a compact sensor, both bulkand Young's moduli are important.

In FIG. 2a, the coating D which is applied directly to the glasswaveguide is soft, such as a rubber or a soft UV curable elastomer,introduced for minimizing microbending losses. For typical thicknesses,these soft elastomers having a very small Young's modulus have verylittle effect on the sensitivity of typical fibers. The outer coating Eis hard, such as a thermoset plastic or a hard UV curable elastomer,introduced for preserving glass strength, protecting fiber from adverseenvironments and facilitating fiber handling. The elastic moduli ofpolymers required to predict the fiber acoustic sensitivity were studiedas a function of frequency and temperature. The frequency andtemperature dependence of the Young's modulus was obtained frommeasurements on bulk samples in the form of rods with diameter 1.5-3.5cm and length 5-15 cm. The bulk modulus was considered to be frequencyindependent and was measured at only one convenient frequency (1 MHz).

For most of the commonly used polymeric coating materials, the elasticmoduli required to predict the fiber acoustic sensitivity are notgenerally known, particularly as a function of frequency andtemperature. The elastic moduli of several commonly used thermoplastics,rubbers, and UV curable elastomers in the temperature range of 0°-35° C.for frequencies 10² to 10⁴ and 1 MHz have been studied. Table II liststhe generic names and the trade names of the polymers studied. Table IIalso lists the Young's and the bulk moduli measured experimentally at 1MHz at three temperatures. The variations of the Young's and bulk moduliare also given in Table II.

                                      TABLE II                                    __________________________________________________________________________    Temperature Dependence Of Young's And                                         Bulk Modulus Of Elastomers At 1 MHz                                                                     Young's                                                                              Varia-                                                                            Bulk   Varia-                                                  Temp.                                                                             Modulus                                                                              tion                                                                              Modulus                                                                              tion                              Generic Name  Trade Name                                                                            °C.                                                                        10.sup.10 dyn/cm.sup.2                                                               %   10.sup.10 dyn/cm.sup.2                                                               %                                 __________________________________________________________________________     (1)                                                                             Polytetra- TEFLON TFE                                                                             0  1.55       2.93                                        fluoroethylene                                                                           Type II 23  1.21   72. 2.61   20.                                                     35  0.900      2.45                                      (2)                                                                             Polytetra- TEFLON TFE                                                                             0  2.20       3.62                                        fluoroethylene                                                                           Type I  27  1.35   85. 2.96   22.                                                     35  1.19       3.10                                      (3)                                                                             Polytetra- TEFLON   0  1.05       3.49                                        fluoroethylene                                                                           FEP 100 26  0.833  29. 2.98   21.                                                     35  0.814      2.88                                      (4)                                                                             polytetra- TEFZEL 280                                                                            0   1.94       4.29                                        fluoroethylene     26  1.59   28. 3.72   18.                                                     35  1.51       3.64                                      (5)                                                                             Polyamide  ZYTEL 101                                                                              0  3.86       6.29                                                           23  3.53   10. 5.98   8.                                                      35  3.52       5.80                                       (6)                                                                            Polyamide  ZYTEL 105                                                                              0  3.94       6.52                                                           22  3.59   12. 5.95   13.                                                     35  3.52       5.75                                      (7)                                                                             Acetal     DELRIN 150                                                                             0  5.11       6.28                                                           22  4.22   28. 5.71   10.                                                     35  4.00       5.82                                      (8)                                                                             Acetal     DELRIN 107                                                                             0  4.98       6.40                                                           22  4.31   49. 5.70   12.                                                     35  3.35       6.15                                      (9)                                                                             Acrylic    LUCITE   0  6.52       6.22                                                           24  5.93   11. 5.81   7.                                                      35  5.85       5.79                                     (10)                                                                             Phenyloxide                                                                              NORYL    0  3.07       4.14                                                   EN265   22  2.95   7.  3.86   8.                                                      35  2.88       3.84                                     (11)                                                                             POLYETHYLENE        0  2.25       3.91                                        Low Density        22  1.92   36. 3.70   12.                                                     35  1.65       3.50                                     (12)                                                                             POLYETHYLENE        0  1.86       3.87                                                           23  2.01   8.  2.97   51.                                                     35  1.86       2.56                                     (13)                                                                             POLYETHYLENE                                                                             ULTRA-   0  2.92       4.55                                        High density                                                                             ETHYLUX 25  2.29   37. 4.26   15.                                                     35  2.13       3.94                                     (14)                                                                             POLYCARBONATE                                                                            LEXAN    0  3.12       4.82                                                   FL 930  22  2.76   17. 4.57   12.                                                     35  2.66       4.30                                     (15)                                                                             POLYPROPYLENE                                                                            PROFAX-  0  4.72       4.76                                                   6823    22  3.49   52. 4.34   19.                                                     35  3.10       3.99                                     (16)                                                                             POLYPROPYLENE                                                                            PROFAX-  0  4.04       4.42                                                   7823    22  2.55   78. 3.83   23.                                                     35  2.27       3.59                                     (17)                                                                             POLYSTYRENE         0  3.97       4.06                                                           25  3.77   5.  3.95   5.                                                      35  3.76       3.85                                     (18)                                                                             POLYSTYRENE                                                                              FOSTARENE-                                                                             0  3.71       3.84                                                   50      22  3.65   5.  3.76   4.                                                      35  3.53       3.70                                     (19)                                                                             POLYSTYRENE                                                                              REXOLITE-                                                                              0  3.79       4.01                                        (Cross-Linked)                                                                           1422    22  3.65   4.  3.82   6.                                                      35  3.65       3.78                                     (20)                                                                             POLYSULFONE                                                                              UDEL     0  3.06       5.08                                                           22  2.93   8.  4.92   6.                                                      35  2.84       4.80                                     (21)                                                                             POLYVINYL  PVC      0  4.50       5.66                                        CHLORIDE           22  4.34   6.  5.41   8.                                                      35  4.25       5.26                                     __________________________________________________________________________

As can be seen from this table, the least Young's modulus variation isobtained with Rexolite (19), polystyrene (17,18), polyvinyl chloride(21), Noryl (10), polysulfone (20), nylon (5,6), and acrylic (9). Theleast bulk modulus dependence is found in polystyrene (17,18), Rexolite(19), polysulfone (20), acrylic (9), Noryl (10), polyvinyl chloride(21), and nylon (5).

In order to obtain the frequency dependence of the elastic moduli of thepolymers, the frequency dependence of the Young's modulus was studied inthe range of 10² -10⁴ Hz at three different temperatures assuming thebulk modulus to be frequency independent. It was found that away fromany transitions the Young's modulus is proportional to the log offrequency, i.e. E=A log₁₀ F+B. The coefficients A and B of the bestlinear fit are listed in Table III. The last three columns give theoverall frequency and temperature variation of the Young's modulus, ther.m.s. fitting error, and the number of experimental points. From TableIII it can be seen that the used equation describes well the Young'smodulus in most polymers in the frequency range of 10² 14 10⁴ Hz.

In general, as it was expected, the Young's modulus decreases astemperature increases or as frequency decreases. Similarly, the bulkmodulus decreases as temperature increases. Table III shows that leasttemperature and frequency variation is obtained with Rexolite (19),polystyrene (17,18), Noryl (10), polycarbonate (14), polysulfone (20),and nylon (6).

                                      TABLE III                                   __________________________________________________________________________    Young's Modulus Frequency Dependence                                                                              Fitting                                                 Temperature                                                                          E = Alog.sub.10 F + B                                                                   Variation                                                                          Error                                                                             No. of                                Elastomer     °C                                                                            A    B    %    %   Points                                __________________________________________________________________________     (1)                                                                             TEFLON TFE  0     .026 1.14      0.8 20                                       Type II    23     .096 0.59 143. 2.3 42                                                  35     .072 0.37      1.8 16                                     (2)                                                                             TEFLON TFE  0     .257 1.42      1.6 11                                       Type I     27     .145 1.09 119. 0.7 12                                                  35     .114 0.89      3.4 17                                     (3)                                                                             TEFLON      0     .093 0.51      2.4 16                                       FEP 100    23     .053 0.55 34.  2.1 16                                                  35     .028 0.60      2.4 15                                     (4)                                                                             TEFZEL      0     .095 1.34      1.3 11                                                  24     .076 1.16 45.  .8  11                                                  35     .099 0.99      2.4 18                                     (5)                                                                             ZYTEL 101   0     .081 3.33      1.4 14                                                  22     .097 2.96 38.  2.3 75                                                  35     .152 2.31      1.8  9                                     (6)                                                                             ZYTEL 105   0     .077 3.10      2.1  9                                                  22     .046 3.09 12.  .7  14                                                  35     .054 2.93      1.5  9                                     (7)                                                                             DELRIN 150  0     .009 3.89      1.7  8                                                  22     .055 3.33 22.  1.4 31                                                  35     .055 3.10      1.3 13                                     (8)                                                                             DELRIN 107  0     .163 3.37      1.3 10                                                  22     .099 3.28 27.  0.3 13                                                  35     .138 2.88      0.8 11                                     (9)                                                                             LUCITE      0     .224 5.10      2.2 36                                                  24     .509 3.79 34.  3.8 27                                                  35     .722 3.04      1.5 12                                    (10)                                                                             NORYL       0     .034 2.65      .8  11                                                  22     .051 2.53 8.   .9  22                                                  35     .032 2.51      1.5 14                                    (11)                                                                             POLYETHYLENE                                                                              0     .001 1.94      1.2 12                                       (Low Density)                                                                            22     .069 1.39 49.  1.3 31                                                  35     .058 1.19      1.3 18                                    (12)                                                                             POLYETHYLENE                                                                              0     .130 0.699     2.57                                                                              12                                       (Low Density)                                                                            22     .096 0.432                                                                              173. 1.1 23                                                  35     .048 0.351     2.7 28                                    (13)                                                                             POLYETHYLENE                                                                              0     .132 2.22      2.9 30                                       (High Density)                                                                           22     .116 1.68 57.  1.9 30                                                  35     .133 1.48      2.0 17                                    (14)                                                                             POLYCARBONATE                                                                             0     .027 2.45      .6  15                                                  22     .047 2.23 10.  .6  26                                                  35     .003 2.33      2.3 15                                    (15)                                                                             POLYPROPYLENE                                                                             0     .324 2.93      .9  12                                       6823       22     .469 0.82 180. 4.2 15                                                  35     .319 0.87      4.7 12                                    (16)                                                                             POLYPROPYLENE                                                                             0     .221 2.52      2.5 13                                       7823       27     .283 0.64 238. 5.6 10                                                  35     .309 0.39      1.9 11                                    (17)                                                                             POLYSTYRENE                                                                               0     .020 3.80      5.4  9                                                  25     .061 3.57 7.   1.3 23                                                  35     .031 3.55      1.0  9                                    (18)                                                                             POLYSTYRENE                                                                               0     .087 3.16      .9  10                                                  22     .037 3.32 7.   .7  14                                                  35     .052 3.17      2.3 27                                    (19)                                                                             REXOLITE    0     .040 3.43      1.3 12                                                  22     .049 3.32 6.   1.3 25                                                  35     .025 3.33      1.9 14                                    (20)                                                                             POLYSULFONE                                                                               0     .131 2.17      3.5 13                                                  22     .069 2.42 11.  0.3 15                                                  35     .032 2.51      0.6 13                                    (21)                                                                             POLYVINYL-  0     .123 3.46      .7  11                                       CHLORIDE   22     .129 3.45 22.  .7  10                                                  35     .225 2.80      .5  13                                    (22)                                                                             HYTREL #7246                                                                              0     .031 1.92      2.3 13                                                  22     .163 1.80 147. 1.1 20                                                  35     .148 0.53      1.5 37                                    (23)                                                                             HYTREL 6346                                                                               0     .285 0.58      3.2 13                                                  24     .187 0.25 176. 1.2 18                                    (24)                                                                             HYTREL 5556                                                                               0     1.088                                                                              1.29      1.5 12                                                  23     .336 2.64 70.  1.2 17                                    (25)                                                                             HYTREL 5526                                                                               0     1.051                                                                              1.89      1.6 18                                                  22     .532 1.94 109. 2.8 20                                                  35     .311 2.30      1.5 12                                    (26)                                                                             HYTREL 4046                                                                              22     1.479                                                                              3.76 56.  2.4 18                                                  35     1.215                                                                              3.78      4.9 14                                    (27)                                                                             UV CURABLE  0     .207 -1.22     7.1  5                                       Hard       26     2.641                                                                              -2.59     7.4  5                                                  35     2.875                                                                              -4.17     8.0  4                                    (28)                                                                             UV CURABLE  0     .644 -1.21     10.2                                                                               6                                       Soft       22     1.324                                                                              -3.22     26.3                                                                               4                                                  35     1.153                                                                              - 3.35                                              (29)                                                                             POLYURETHANE                                                                              0     .797 0.85      1.3 28                                       EN-9       22     .131 0.52 512. 4.6 14                                                  35     .110 0.44      3.0 14                                    (30)                                                                             POLYURETHANE-                                                                             0     .674 0.44      2.6 12                                       3130       22     .238 0.08 771. 3.7 14                                                  35     .165 0.03      10.3                                                                              14                                    (31)                                                                             POLYURETHANE-                                                                             0     1.961                                                                              1.52      3.8 10                                       PR 1592    22     .976 1.02 571. 4.5 12                                                  35     .468 0.46      3.8 12                                    __________________________________________________________________________

Using the hydrostatic analysis the acoustic sensitivity of fibers coatedwith various polymers was obtained from the bulk and Young's modulilisted in Tables II and III. The results are given in Table IV. In thesecalculations, the fiber parameters were taken from Table I and the fibero.d. was taken to be 0.35 mm, a "thin" fiber, 1 mm, as used in thetested sensors, and 4 mm, a "thick" coating case. The results for fiberswith thick coatings maximum sensitivity is obtained with PolyurethaneConap (29) and Uralite 3130 (30), and Teflon TFE II (1) due to their lowbulk modulus. For fibers with typical thickness coatings, however,maximum sensitivity is obtained with polystyrenes (17,18) and Rexolite(19) due primarily to their high Young's modulus, and with Teflon TFE(1,2). The smallest temperature and frequency dependence is obtainedwith polystyrene (18), polyvinyl chloride (21), Lucite (9), and Noryl(10). Small dependence is also obtained with Delvin (7) and Zytel (101). On the other hand, high dependence is obtained with the soft UVcurable elastomer due to the big frequency and temperature dependence ofthe Young's modulus.

                                      TABLE IV                                    __________________________________________________________________________    Acoustic Sensitivity of Fibers Coated with Various Elastomers                 [Δφ/φ ΔP (10.sup.-12 /(dyn/cm.sup.2))]                                      Typical Fibers Fiber          Thick Fibers                                    (o.d. 0.35 mm)                                                                          Total                                                                              (o.d. 1 mm)                                                                             Total                                                                              (o.d. 4                                                                                 Total                             Temp                                                                              Frequency (Hz)                                                                          Variation                                                                          Frequency (Hz)                                                                          Variation                                                                          Frequency                                                                               Variation           Polymer       (°C.)                                                                      10.sup.2                                                                          10.sup.3                                                                         10.sup.4                                                                         (%)  10.sup.2                                                                          10.sup.3                                                                         10.sup.4                                                                         (%)  10.sup.2                                                                          10.sup.3                                                                         10.sup.4                                                                         (%)                 __________________________________________________________________________     (1)                                                                             TEFLON TFE  0  1.69                                                                              1.72                                                                             1.74    6.23                                                                              6.25                                                                             6.27    8.38                                                                              8.37                                                                             8.35                      Type II    23  1.36                                                                              1.48                                                                             1.60                                                                             39   6.10                                                                              6.33                                                                             6.56                                                                             16   9.31                                                                              9.34                                                                             9.36                                                                             15                                35  1.06                                                                              1.17                                                                             1.27    5.48   6.09    9.81                                                                              9.84                                                                             9.86                    (2)                                                                             TEFLON TFE  0  1.99                                                                              2.18                                                                             2.37    5.62                                                                              5.79                                                                             5.88    6.79                                                                              6.81                                                                             6.83                      Type I     27  1.76                                                                              2.54                                                                             3.31                                                                             51   5.99                                                                              6.12                                                                             6.25                                                                             10   7.77                                                                              7.78                                                                             7.78                                                                             15                                35  1.60                                                                              1.69                                                                             1.77    5.85                                                                              5.97                                                                             6.09    7.87                                                                              7.91                                                                             7.95                    (3)                                                                             TEFLON      0  0.98                                                                              1.07                                                                             1.16    4.27                                                                              4.49                                                                             4.71    6.73                                                                              6.81                                                                             6.88                      FEP 100    23  1.08                                                                              1.13                                                                             1.18                                                                             17   4.99                                                                              5.09                                                                             5.19                                                                             19   8.04                                                                              8.01                                                                             7.97                                                                             19                                35  1.11                                                                              1.14                                                                             1.17    5.12                                                                              5.21                                                                             5.29    8.28                                                                              8.31                                                                             8.33                    (4)                                                                             TEFZEL      0  1.43                                                                              1.50                                                                             1.57    4.43                                                                              4.52                                                                             4.60    5.59                                                                              5.63                                                                             5.67                      280        24  1.47                                                                              1.52                                                                             1.56                                                                             12   5.00                                                                              5.05                                                                             5.09                                                                             15   6.56                                                                              6.58                                                                             6.50                                                                             17                                35  1.38                                                                              1.48                                                                             1.57    4.91                                                                              5.06                                                                             5.20    6.59                                                                              6.66                                                                             6.73                    (5)                                                                             ZYTEL       0  1.79                                                                              1.83                                                                             1.86    3.48                                                                              3.51                                                                             3.54    3.84                                                                              3.86                                                                             3.87                      101        22  1.77                                                                              1.80                                                                             1.83                                                                             11   3.65                                                                              3.66                                                                             3.66                                                                              6   4.07                                                                              4.06                                                                             4.04                                                                              8                                35  1.66                                                                              1.69                                                                             1.71    3.64                                                                              3.68                                                                             3.72    4.17                                                                              4.18                                                                             4.19                    (6)                                                                             ZYTEL       0  1.66                                                                              1.69                                                                             1.71    3.33                                                                              3.35                                                                             3.36    3.70                                                                              3.70                                                                             3.70                      105        22  1.78                                                                              1.80                                                                             1.82                                                                              9   3.66                                                                              3.68                                                                             3.70                                                                             10   4.08                                                                              3.10                                                                             4.11                                                                             10                                35  1.73                                                                              1.75                                                                             1.77    3.63                                                                              3.64                                                                             3.65    4.07                                                                              4.07                                                                             4.07                    (7)                                                                             DELRIN      0  1.95                                                                              1.95                                                                             1.94    3.58                                                                              3.57                                                                             3.56    3.89                                                                              3.88                                                                             3.87                      150        22  1.97                                                                              1.99                                                                             2.00                                                                              9   3.89                                                                              3.90                                                                             3.90                                                                              9   4.30                                                                              4.30                                                                             4.29                                                                             10                                35  1.83                                                                              1.85                                                                             1.87    3.76                                                                              3.77                                                                             3.77    4.17                                                                              4.17                                                                             4.17                    (8)                                                                             DELRIN      0  1.83                                                                              1.89                                                                             1.95    3.46                                                                              3.49                                                                             3.52    3.78                                                                              3.80                                                                             3.82                      107        22  1.98                                                                              2.02                                                                             2.06                                                                             17   3.89                                                                              3.91                                                                             3.93                                                                             12   4.29                                                                              4.30                                                                             4.31                                                                             12                                35  1.71                                                                              1.76                                                                             1.80    3.50                                                                              3.52                                                                             3.54    3.91                                                                              3.91                                                                             3.91                    (9)                                                                             LUCITE      0  2.45                                                                              2.50                                                                             2.55    3.67                                                                              3.73                                                                             3.78    3.97                                                                              3.97                                                                             3.96                                 24  2.39                                                                              2.54                                                                             2.68                                                                             36   3.97                                                                              4.01                                                                             4.05                                                                             14   4.24                                                                              4.25                                                                             4.25                                                                              8                                35  1.71                                                                              1.76                                                                             1.80    3.50                                                                              3.52                                                                             3.54    3.91                                                                              3.91                                                                             3.91                   (10)                                                                             NORYL       0  2.24                                                                              2.27                                                                             2.29    5.18                                                                              5.20                                                                             5.22    5.91                                                                              5.92                                                                             5.93                                 22  2.36                                                                              2.39                                                                             2.41                                                                              6   5.59                                                                              5.60                                                                             5.61                                                                              8   6.42                                                                              6.41                                                                             6.40                                                                              8                                35  2.31                                                                              2.34                                                                             2.37    5.55                                                                              5.59                                                                             5.62    6.39                                                                              6.42                                                                             6.45                   (11)                                                                             POLYETHYLENE                                                                              0  1.87                                                                              1.87                                                                             1.87    5.21                                                                              5.21                                                                             5.21    6.29                                                                              6.29                                                                             6.29                      (Low Density)                                                                            22  1.65                                                                              1.70                                                                             1.75                                                                             18   5.22                                                                              5.36                                                                             5.45                                                                              4   6.60   6.58                                                                             10                                35  1.54                                                                              1.60                                                                             1.65    5.27                                                                              5.36                                                                             5.45    6.90                                                                              6.96                                                                             7.01                   (12)                                                                             POLYETHYLENE                                                                              0  1.13                                                                              1.23                                                                             1.33    4.32                                                                              4.49                                                                             4.66    6.14                                                                              6.16                                                                             6.18                      (Low Density)                                                                            22  1.07                                                                              1.18                                                                             1.29                                                                             43   5.04                                                                              5.29                                                                             5.54                                                                             22   8.33                                                                              8.32                                                                             8.31                                                                             26                                35  1.72                                                                              1.81                                                                             1.89    5.03                                                                              5.11                                                                             5.18    6.18                                                                              6.20                                                                             6.22                   (13)                                                                             POLYETHYLENE                                                                              0  1.93                                                                              2.01                                                                             2.08    4.66                                                                              4.71                                                                             4.75    5.39                                                                              5.40                                                                             5.41                      (High Density)                                                                           22  1.71                                                                              1.78                                                                             1.85                                                                             18   4.75                                                                              4.81                                                                             4.86                                                                             10   5.74                                                                              5.75                                                                             5.76                                                                             13                                35  1.72                                                                              1.81                                                                             1.89    5.03                                                                              5.11                                                                             5.18    6.18                                                                              6.20                                                                             6.22                   (14)                                                                             POLYCARBONATE                                                                             0  1.83                                                                              1.83                                                                             1.83    4.36                                                                              4.36                                                                             4.36    5.04                                                                              5.04                                                                             5.04                                 22  1.86                                                                              1.85                                                                             1.84                                                                              6   4.38                                                                              4.49                                                                             4.59                                                                             11   5.04                                                                              5.21                                                                             5.37                                                                             12                                35  1.88                                                                              1.92                                                                             1.96    4.59                                                                              4.75                                                                             4.90    5.33                                                                              5.53                                                                             5.73                   (15)                                                                             POLYPROPYLENE                                                                             0  2.38                                                                              2.52                                                                             2.66    4.69                                                                              4.75                                                                             4.80    5.16                                                                              5.18                                                                             5.20                      6823       22  1.59                                                                              1.55                                                                             1.52                                                                             43   4.59                                                                              4.77                                                                             4.94                                                                             12   5.63                                                                              5.64                                                                             5.64                                                                             16                                35  2.13                                                                              2.05                                                                             1.97    4.80                                                                              5.00                                                                             5.20    6.08                                                                              6.12                                                                             6.17                   (16)                                                                             POLYPROPYLENE                                                                             0  2.25                                                                              2.36                                                                             2.47    4.95                                                                              4.99                                                                             5.03    5.59                                                                              5.59                                                                             5.58                      7823       27  1.36                                                                              1.48                                                                             1.59                                                                             49   4.76                                                                              4.98                                                                             5.20                                                                             13   6.35                                                                              6.37                                                                             6.38                                                                             18                                35  1.25                                                                              1.52                                                                             1.78    4.80                                                                              5.14                                                                             5.48    6.73                                                                              6.79                                                                             6.84                   (17)                                                                             POLYSTYRENE                                                                               0  2.91                                                                              2.92                                                                             2.92    5.61                                                                              5.60                                                                             5.59    6.13                                                                              6.13                                                                             6.12                                 25  2.91                                                                              2.94                                                                             2.97                                                                              2   5.73                                                                              5.75                                                                             5.76                                                                              5   6.30                                                                              6.30                                                                             6.30                                                                              5                                35  2.93                                                                              2.95                                                                             2.97    5.85                                                                              5.87                                                                             5.88    6.44                                                                              6.45                                                                             6.46                   (18)                                                                             POLYSTYRENE                                                                               0  2.77                                                                              2.79                                                                             2.81    5.79                                                                              5.74                                                                             5.68    6.43                                                                              6.36                                                                             6.28                                 22  2.87                                                                              2.90                                                                             2.92                                                                              7   5.94                                                                              5.96                                                                             5.98                                                                              5   6.59                                                                              6.61                                                                             6.62                                                                              6                                35  2.84                                                                              2.98                                                                             2.91    6.00                                                                              6.02                                                                             6.04    6.70                                                                              6.70                                                                             6.70                   (19)                                                                             REXOLITE    0  2.77                                                                              2.79                                                                             2.81    5.60                                                                              5.61                                                                             5.61    6.19                                                                              6.19                                                                             6.18                                 22  2.84                                                                              2.88                                                                             2.91                                                                              5   5.85                                                                              5.88                                                                             5.90                                                                              5   6.49                                                                              6.51                                                                             6.52                                                                              6                                35  2.85                                                                              2.87                                                                             2.88    5.91                                                                              5.92                                                                             5.92    6.57                                                                              6.57                                                                             6.56                   (20)                                                                             POLYSULFONE                                                                               0  1.74   1.84    4.15   4.19    4.80   4.79                                 22  1.84   1.89                                                                             10   4.32   4.33                                                                              7   4.98   4.94                                                                              6                                35  1.88   1.93    4.41   4.46    5.07   5.11                   (21)                                                                             POLYVINYL-  0  2.07   2.16    3.94   3.97    4.31   4.32                      CHLORIDE   22  2.16   2.26                                                                              9   4.13   4.17                                                                              7   4.52   4.53                                                                              8                                35  2.03   2.22    4.18   4.25    4.65   4.66                   (22)                                                                             HYTREL      0  1.49   1.52    3.97   4.00    4.76   4.76                      7246       22  1.12   1.34                                                                             37   3.93   4.22                                                                              9   5.33   5.39                                                                             16                                35  0.96   1.17    3.82   4.16    5.66   5.65                   (23)                                                                             HYTREL      0  1.03   1.34    3.49   3.82    4.70   4.70                      6346       24  0.76   1.01                                                                             43   3.18   3.69                                                                             17   5.19   5.18                                                                              9                                 0                                                              (24)                                                                             HYTREL      0  3.12           6.41           7.11                             5556       23                                                                            35                                                              (29)                                                                             POLYURETHANE                                                                              0  3.17   4.59    8.43   8.93    9.82   9.76                      CONAP      22  1.48   1.83                                                                             69   6.72   7.38                                                                             25   10.28  10.33                                                                            15                                35  1.43   1.77    7.05   7.82    11.45  11.52                  (30)                                                                             POLYURETHANE-                                                                             0  1.98   2.98    5.84   6.48    7.16   7.27                      3130       22  1.05   1.63                                                                             72   5.20   6.49                                                                             31   8.94   9.11                                                                             26                                35  0.83   1.29    4.46   6.06    9.31   9.66                   (31)                                                                             POLYURETHANE-                                                                             0  3.84   5.17    6.20   6.47    6.56   6.61                      1592       22  2.92   4.09                                                                             45   6.58   7.01                                                                             16   7.44   7.50                                                                             46                                35  4.46   2.84    9.82   7.43    12.08  8.71                   __________________________________________________________________________

In order to increase the acoustic sensitivity and to minimizeacceleration effects, fibers with substantially thicker coatings thanthose of the standard communication fibers (0.5-0.6 mm o.d.) must beused. The first thick fibers were coated with nylon up to 1 mm o.d. Itwas found, however, that the acoustic sensitivity of these thick fibers,even though initially was high and in agreement with that predictedanalytically (Table IV), deteriorated fast as the fibers were immersedin water (or a hydrocarbon solvent). After careful examination, it wasfound that the thick nylon coating had poor bonding to silicon and bigair gaps were in the nylon-silicone interface. When such fibers areimmersed in liquids, axial stresses can not communicate from the outercoating to the inner coating. Such fibers can be used only as referenceor lead fibers since they are desensitized to pressure and accelerationat the same time. The poor interface between nylon and silicone isformed during the extrusion process when the outer surface of the thicknylon is cooled first, becomes rigid and pulls the inner soft nylon awayfrom the primary coating.

Several solutions to the thick coating delamination problem arepossible. Chemical bonding between the primary and the outer coatingwill certainly result in strong bonding. Also, applying the outercoating in multipasses is expected to enhance fiber bonding. In additionto silicone, U.V. curable polymers are expected to have good bondingwith glass, contrary to others used in communication applications whichhave poor bonding to glass in order to facilitate mechanical removal ofthe coating in the field.

In order to obtain bonding between the outer thick coating and silicone,a modified nylon was used containing 10% silicone molecules cross linkedto nylon molecules. Such modified nylon is expected to have chemicalbonding with the primary silicone copating when the silicone moleculeslink together.

Table V, column 1, lists the most promising polymers as fiber coatingsfor interferometric sensor. The second column shows the calculatedacoustic sensitivity of single mode fibers (Table I) coated with thesepolymers up to 1 mm o.d. The third column shows the overall variation ofthe acoustic sensitivity in the temperature range of 0°-35° C. and forfrequencies 10² -10⁴ Hz. The fourth and fifth columns show the predictedminimum detectable pressure when the reference fiber of theinterferometric sensor is coated with a 0.35 mm o.d. nylon (zytel 101)and a 110 o.d. aluminum, respectively. In these calculations it isassumed that the sensing fiber is 30 m and the minimum detectable phasemodulation is 10⁻⁶ radians for 1 Hz bandwidth (Eq. (1)). Table V showsthat the highest sensitivity is obtained for TFE (due primarily to itslow bulk modulus) with, however, significant temperature and frequencyvariation. Rexolite (a cross-linked styrene polymer) and TFA-340 alsogive high sensitivity and very small variation. Noryl gives highsensitivity (due to its high bulk modulus) and the smallest temperatureand frequency variation. In general, good sensitivity is obtained withall Teflons as a result of their low bulk modulus.

                                      TABLE V                                     __________________________________________________________________________    Optimized Coatings                                                                                Temp. and                                                                           P.sub.min P.sub.min                                                     Frequency                                                                           (Ref. Fiber                                                                             (Ref. Fiber                               Coating    P        Variation                                                                           Nylon:0.35 mm o.d.                                                                      Al, 110 μm                             o.d.       (10.sup.-12 /dyn/cm.sup.2)                                                             %     dB ref 1μ Pa                                                                         dB ref 1μ Pa                           __________________________________________________________________________    TEFLON.sup.1                                                                             6.13     15    37.2      34.3                                      TFE                                                                           POLYSTYRENE                                                                              5.96     5     37.5      34.5                                      REXOLITE   5.87     5     37.7      34.7                                      TEFLON     5.87     6     37.7      34.7                                      TFA-340                                                                       NORYL      5.59     4     38.3      35.1                                      POLYETHYLENE                                                                             5.26     6     39.1      35.1                                      (Low Density)                                                                 TEFLON     5.12     18    39.5      35.9                                      FEP 100                                                                       TEFZEL     5.05     13    39.6      36.0                                      280                                                                           POLYEHTYLENE                                                                             4.79     9     40.4      36.5                                      (High DENSITY)                                                                POLYPROPYLENE                                                                            4.79     16    40.4      36.5                                      6823                                                                          HYTREL     4.09     16    42.7      37.9                                      7246                                                                          NYLON      3.65     3     44.6      38.9                                      101                                                                           __________________________________________________________________________

The foregoing has described an optical fiber sensor having optimizedcoatings for providing maximum acoustic pressure sensitivity. As wasmentioned above, high sensitivity is obtained with fiber coatingpolymers having low bulk modulus and/or high Young's modulus. With highYoung's modulus polymers, high sensitivity can be obtained with arelatively thin coating, which is highly desirable when there isrestriction in space.

Obviously, numerous modifications and variations of the presentinvention are possible in light of the above teachings. For example anymedium having high optical transparency, e.g. glass or plastic, could beused as the core and cladding layers in the disclosed fiber. It istherefore to be understood that within the scope of the appended claims,the invention may be practiced otherwise than as specifically describedherein.

What is claimed as new and is desired to be secured by Letters Patent ofthe United States is:
 1. An optical fiber sensor with maximum pressuresensitivity, Δφ/φΔP, made by:selecting a material for the fiber core andsubstrate; measuring the Young's modulus, Poisson's Ratio, elastopticcoefficients, and refractive index of said selected core and substratematerials; selecting a diameter for said core and substrate; identifyingcandidate materials for a soft coating, and a hard coating to be appliedas layers over said core and substrate; measuring the Young's modulusand the Poisson's Ratio for each said candidate material; selectingmaterials for said soft coating, and hard coating from said candidatematerials; selecting a minimum and a maximum thickness for said softcoating, and hard coating layers; setting said soft and hard coatingthicknesses to said minimum thicknesses; equating the polar stresses insaid core, substrate, soft, and hard coating layers to the matrixproduct of Lame parameters and the core cylindrical strains; solvingsaid matrix product to obtain equations expressing said cylindricalstrains in terms of the constants U₀ ^(i), U₁ ^(i), W₀ ^(i) ; applyingboundary conditions to determine values for said constants; calculatingsaid cylindrical strains from said constant values; calculating saidpressure sensitivity from said cylindrical strains, said refractiveindex, and said elasto-optic coefficients; iterating above saidcalculations from said minimum to said maximum thickness of said hardcoating; incrementally increasing the thickness of said soft layer fromsaid minimum to said maximum thickness; repeating above saidcalculations for each said incremented thickness; identifying thecombination of thickness for said soft coating and hard coating thatgenerates a maximum pressure sensitivity for the selected coatingmaterials; repeating said calculations and iterations to identify amaximum pressure sensitivity for each said candidate material; selectingthe greatest maximum pressure sensitivity from among said identifiedmaximum sensitivities; identifying said soft and hard coating materialsand said thicknesses that generated said greatest maximum pressuresensitivity; fabricating a fiber optic sensor fiber using said selectedcore and substrate material, said identified soft and hard coatingmaterials, of said identified thicknesses.
 2. An optical fiber sensorwith maximum pressure sensitivity per unit of cross sectional area,Δφ/φΔP * 1/r², made by:selecting a material for the fiber core andsubstrate; measuring the Young's modulus, Poisson's Ratio, elasto-opticcoefficients, and refractive index of said selected core and substratematerials; selecting a diameter for said core and substrate; identifyingcandidate materials for a soft coating, and a hard coating to be appliedas layers over said core and substrate; measuring the Young's modulusand the Poisson's Ratio for each said candidate material; selectingmaterials for said soft coating, and hard coating from said candidatematerials; selecting a minimum and a maximum thickness for said softcoating and hard coating layers, setting said soft and hard coatingthicknesses to said minimum thicknesses; equating the polar stresses insaid core, substrate, soft, and hard coating layers to the matrixproduct of Lame parameters and the core cylindrical strains; solvingsaid matrix product to obtain equations expressing said cylindricalstrains in terms of the constants U₀ ^(i), U₁ ^(i), W₀ ^(i) ; applyingboundary conditions to determine values for said constants; calculatingsaid cylindrical strains from said constant values; calculating saidpressure sensitivity per unit cross sectional area from said cylindricalstrains, said refractive index, and said elasto-optic coefficients;dividing said pressure sensitivity by the square of the radius to theouter surface of said fiber; iterating above said calculations from saidminimum to said maximum thickness of said hard coating; incrementallyincreasing the thickness of said soft layer from said minimum to saidmaximum thickness; repeating above said calculations for each saidincremented thickness; identifying the combination of thickness for saidsoft coating and hard coating that generates a maximum pressuresensitivity for the selected coating materials; repeating saidcalculations and iterations to identify a maximum pressure sensitivityfor each said candidate material; selecting the greatest maximumpressure sensitivity from among said identified maximum sensitivities;identifying said soft and hard coating materials and said thicknessesthat generated said greatest maximum pressure sensitivity per unit crosssectional area; fabricating a fiber optic sensor fiber using saidselected core and substrate material, said identified soft and hardcoating materials, of said identified thicknesses.